Question

The circumference of a circle is 100 cm. The side of a square inscribed in the circle is :

• A. \(50\sqrt{2} \text{ cm}\)
• B. \(\frac{100}{\pi} \text{ cm}\)
• C. \(\frac{50\sqrt{2}}{\pi} \text{ cm}\)
• D. \(\frac{100\sqrt{2}}{\pi} \text{ cm}\)

Hint:

1. Find radius (\(r\)) from circumference (\(C\)): \(C = 2\pi r \implies 100 = 2\pi r \implies r = \frac{50}{\pi}\). 2. Diameter of circle (\(D\)): \(D = 2r = 2 \times \frac{50}{\pi} = \frac{100}{\pi}\). 3. For an inscribed square, diagonal of square = diameter of circle. So, diagonal of square (\(d_{sq}\)) = \(\frac{100}{\pi}\). 4. For a square with side \(s\), \(d_{sq} = s\sqrt{2}\). So, \(s\sqrt{2} = \frac{100}{\pi}\). \(s = \frac{100}{\pi\sqrt{2}}\). 5. Rationalize: \(s = \frac{100\sqrt{2}}{2\pi} = \frac{50\sqrt{2}}{\pi}\).

Answer 1

The Correct Option is C

Concept:
Circumference of a circle (\(C\)) = \(2\pi r\), where \(r\) is the radius.
When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle.
For a square with side \(s\), its diagonal \(d_{sq} = s\sqrt{2}\).
Diameter of a circle (\(D_{circ}\)) = \(2r\). Step 1: Find the radius of the circle Given circumference \(C = 100\) cm. We know \(C = 2\pi r\). So, \(100 = 2\pi r\). Solve for \(r\): \[ r = \frac{100}{2\pi} = \frac{50}{\pi} \text{ cm} \] Step 2: Relate the diagonal of the inscribed square to the diameter of the circle When a square is inscribed in a circle, its vertices lie on the circle. The diagonal of this square is equal to the diameter of the circle. Diameter of the circle, \(D_{circ} = 2r\). Substitute \(r = \frac{50}{\pi}\): \[ D_{circ} = 2 \times \frac{50}{\pi} = \frac{100}{\pi} \text{ cm} \] So, the diagonal of the inscribed square, \(d_{sq} = D_{circ} = \frac{100}{\pi}\) cm. Step 3: Find the side of the square Let the side of the inscribed square be \(s\). The diagonal of a square is related to its side by \(d_{sq} = s\sqrt{2}\). We have \(d_{sq} = \frac{100}{\pi}\). So, \(s\sqrt{2} = \frac{100}{\pi}\). Solve for \(s\): \[ s = \frac{100}{\pi\sqrt{2}} \] To rationalize the denominator (optional but good practice, and helps match options): multiply numerator and denominator by \(\sqrt{2}\). \[ s = \frac{100\sqrt{2}}{\pi\sqrt{2}\sqrt{2}} = \frac{100\sqrt{2}}{\pi \times 2} \] \[ s = \frac{50\sqrt{2}}{\pi} \text{ cm} \] The side of the square inscribed in the circle is \( \frac{50\sqrt{2}}{\pi} \) cm.